A Few Fractals

by Nicholas Mee on January 30, 2013

Following my previous post I have had a number of requests for images of fractals, so here goes.

Fractals are characterised by their property of self-similarity, they are both similar to themselves in different regions and similar to themselves on different length scales, so that an enlargement of a fractal looks the same as the original.

North Britain

A Trip to the Seaside

The notion of fractals was popularized by the Polish mathematician Benoit Mandelbrot (1924-2010) in his book The Fractal Geometry of Nature. Mandelbrot pointed out that many naturally occurring structures are best described in terms of fractals. These include coastlines, trees, clouds, cauliflowers and many other objects. Indeed, any objects that look the same from a distance as when we zoom in to take a closer look. For instance, a small twig often looks just like a miniature tree.

One of Mandelbrot’s favourite examples was the coastline of Britain. As Mandelbrot noted, it is very wiggly. And if we zoom in and take a closer look these wiggles are not ironed out. We just see more smaller wiggles. So that the closer we look the longer the coastline appears to be.

Blowing Your Own Gasket

Mandelbrot was not the first to explore the properties of fractals. Another Polish mathematician Waclaw Sierpinski (1882-1969) is known for a mathematical procedure for generating a fractal structure known as the Sierpinski gasket. Sierpinski’s method of construction was to start with an equilateral triangle, then remove the central triangle to leave three smaller triangles. Each of these triangles is a smaller version of the original triangle. The central triangle is then removed from each of the smaller triangles to leave even smaller equilateral triangles. If this process were continued indefinitely the result would be the Sierpinski gasket, which is composed of a dust of disconnected points.

A similar procedure may be applied to a regular tetrahedron. Removing an octahedron from the centre of the tetrahedron leaves four smaller tetrahedra. Octahedra may then be removed from each of these smaller tetrahedra. Continuing indefinitely produces a tetrahedral Sierpinski gasket, as shown in the animation below.

Julia’s Dream

Another mathematician who took an early interest in fractals was the Frenchman Gaston Julia (1893-1978). He studied the convergence properties of sequences of numbers.These sequences have the following form: take a number c, then for each complex value of x the sequence consists of the terms x(1) = x² + c, x(2) = x(1)² + c, x(3) = x(2)² + c, and so on, each term being equal to the square of the previous term plus c. For some starting numbers x the terms will increase in size indefinitely and the sequence will diverge, whilst for other starting numbers the terms do not increase beyond some finite number. Julia was interested in those values of x that produce sequences that remain finite. These values of x constitute the Julia set of c.

This might sound rather technical, but it is the sort of procedure that a computer is ideally suited to performing and the end results look rather attractive. It is only since computers became commonplace that mathematicians have been able to explore these objects in detail.

By generating a sequence of Julia sets for values of the parameter c corresponding to points on a closed loop it is possible to create animations such as the ones shown here. I originally produced these animations some years ago for the Fractal Geometry section of my CD-ROM Art and Mathematics.

The animations illustrate just how complicated the structure of the Julia sets can be and also how their structure is dramatically dependent on the value of the constant c. The Julia sets fall into two classes: they are either completely connected together or they consist of a dust of isolated points.

As a non mathematician I found the following instructions above confusing:
” Sierpinski’s method of construction was to start with an equilateral triangle, then remove the central triangle to leave three smaller triangles”
Where in an equilateral triangle is the ‘central triangle’? I’ve always understood such a triangle was simply one with three equal sides and equal internal angles. How do I find/calculate this ‘central triangle’?
Sorry if I’m wasting your time.
Thanks.

Sorry about that. What I wrote was a bit cryptic. If you take an equilateral triangle and connect the midpoints of each edge, the original triangle is divided into four smaller equilateral triangles. What I meant by the central triangle is the one at the centre of these four smaller triangle i.e. the one whose edges connect the points at the middle of each edge of the original triangle.

The square root of the number -1 (minus one) does not exist within the real numbers, so mathematicians call this number i. Multiples of the number i are called imaginary numbers, which is perhaps an unfortunate name because all numbers are abstract concepts. Anyway, mathematicians extend the real number system to form the complex number system by adding together real numbers and imaginary numbers, so a complex number consists of two pieces – the real part and the imaginary part. For instance, 7.5 + 3.2i is a complex number, it has a real part 7.5 and an imaginary part 3.2i. Complex numbers can be added and multiplied consistently, just like real numbers, as long as we remember that i squared is equal to -1. It is often very convenient to plot complex numbers as points in a plane, with the two parts of the complex number represented as the x and y components of the position of the point. This is why the Mandelbrot set and Julia sets can be depicted as two dimensional images. The points representing the complex numbers are plotted in different colours depending on the convergence properties of the number sequences beginning with that number.

Hi Nicholas, I have read alot about fractals. This triangle is the most wonderful fractal I have ever seen! I was wondering if you think the positions for planets and stars make giant fractals, just a crazy thought
godbless
Eric